Alternatives to Spherical Microphone arrays: Hybrid

Geometries

Aastha Gupta&

Prof. Thushara AbhayapalaApplied Signal Processing

CECS

To be presented at ICASSP, 20-24 April 2009, Taipei, Taiwan

Outline

• Spherical harmonic analysis of wavefields• Spherical microphone arrays and limitations• Theory of Non-spherical (Hybrid) arrays• Combination of Circular Arrays• Conclusions

Spherical Coordinates

• : Elevation• : Azimuth • r: Radial Distance

Spherical Harmonics

Wave Propagation

• Wavefields/ soundfields are governed by the wave equation.

• Homogeneous fields. They could be due to scattering, diffraction, and refraction.

• Basic solution can act as a set of building blocks

Modal Analysis

Arbitrary Soundfield – General Solution

Spherical Microphone arrays

• Spherical microphone arrays capture soundfield on a surface of a sphere.

• Natural choice for harmonic decomposition.• Open Sphere [Abhayapala & Ward ICASSP 02]• Rigid Sphere [Meyer & Elko, ICASSP 02].• Bessel zeros are a problem in open spheres.• Rigid spheres are less practical for low

frequencies.• Strict orthogonality condition on sensor

locations.

Problem

• Spherical harmonic decomposition of wavefields/soundfield is a great way to solve difficult array signal processing problems.

• How can we estimate spherical harmonics from an array of sensors?

• What are the alternatives to spherical arrays?

Circular Microphone Arrays

Spherical Harmoni

cs

Let be the soundfield on a circle at

Hybrid Arrays

We multiply by and integrate with respect to over to get

where

Circular harmonic Decomposition

Left hand side of this equation is a weighted sum of soundfield coefficients for a given . It can be evaluated for where the truncation number is dependent on the radius of the circle.We show how to extract from a number of carefully placed circular arrays

Sampling of Circles

In practice, we can not obtain soundfield at every point on these circles. Thus, needs samplingAccording to Shannon's sampling theorem for periodic functions, can be reconstructed by its samples over with at least samples . We approximate the integral in by a summation:

are the number of sampling points on the circle .

Least Squares Suppose our goal is to design a Nth order microphone array to estimate

(N + 1)^2 spherical harmonic coefficients. By placing Q ≥ (N + 1) circles of microphones on planes given by (rq, θq), q =1, . . . ,Q, for a specific m, we have

where

The harmonic coefficients can be calculated by solving the simultaneous system of equations or evaluating a valid Moore-Penrose inverse of the matrix

Legendre Properties

Bessel Properties

Infinite summation can be truncated by using properties of Bessel functions:

Number of Coefficients

Combination of Circles

• Consider two circles placed at and where 0 ≤ ≤ .• That is one circle above the x-y plane and the second circle below the x-y plane but equal distance rq from the origin• The circular harmonics of the soundfield on the circle on or above the x-y plane are given by

Above xy plane

Below xy plane

Circular Harmonic combination

• Right hand side is a weighted sum of coefficients for a specific

• For l=0 the sum only consists of a weighted sum of with n is even.

• For l=1 the sum only consists of a weighted sum of with n is odd.

Findings so far..

•Thus, we can separate odd and even sphericalharmonics from the measurement of soundfield on two

circlesplaced on equal distance above and below the x-y plane.

•This is a powerful result, which we can useto extract spherical harmonics from soundfield

measurementson carefully placed pairs of circles.

Odd Coefficients

There are specific patterns of the normalized associated Legendre function when n−|m| =1, 3, 5... There are number of different range of elevation angles we can choose for θq. Note that θq could be same for all q or a group of values.

For a Nth order system, there are N(N +1)/2 odd spherical harmonic coefficients from total of (N +1)2 coefficients. We use N (for N odd) or N − 1 (for N even) pairs of of circular microphone arrays. We choose the radii of these circles as

Guidelines to choose systematicallysuch that is always non singular:

Findings

•With this choice, the soundfield at frequency k on a circlewith rq is order limited to due tothe properties of Bessel functions. This property limits thehigher order components of the soundfield present at a particularradius rq. Also, the lower order components are guaranteedto be present due to the choice of radii in (14) whichavoids the Bessel zeros.

• Thus, selecting rq and θq from the legendre and Bessel plots, we can guarantee that is non singular.

Normalised Legendre function-odd

Even Coefficients Suppose, we have selected Q pairs of such that when is even.

We have following guidelines to choose systematically such that is always non singular:

As in the case of odd coefficients, we can choose range ofvalues for θq, which plots for even.• Note that on the x-y plane (θ = π/2), all even associate Legendrefunctions are non zero. Thus, placing circles on the x-yplane seems to be an obvious choice to estimate even coefficients, where we do not need pairs of circles.

Normalised Legendre Function-even

Findings Depending on our choice, we can design different

array configurations, which will be capable of estimating spherical harmonic coefficients.

For a Nth order system, we place N/2 (N even) or (N+1)/2 (N odd) circles on the x-y plane. We choose the radii of these circles based on the bessel plots

Simulations-5th Order System We first place four circular arrays (two pairs) with

11, 11, 7 and 7 microphones at (4/ko, π/3), (4/ko, π − π/3), (5/ko, π/6), and (4/ko, π − π/6). Then we place a pair of microphones at (5/ko, 0) and (5/ko, π).

This sub array consists of 38 microphones are designed to calculate all odd spherical harmonics up to the 5th order (total of 15 coefficients).

We place three circular arrays on the x-y plane together with a single microphone at the origin to complete the design. We have 7, 11, and 13 microphones in three arrays on x-y plane at radial distances 2/ko, 4/ko, and 5/ko, respectively.

Simulations

Real part of the estimated harmonic coefficient α54 for aplane wave sweeping over entire 3D space: (a) Theoretical pattern(b), (c), (d) are at frequencies 3000, 4500 and 6000Hz, respectively, and all at SNR= 40dB

•Test Octave - 3000Hz to 6000Hz (kℓ = 55.44)

•40dB signal to noise ratio (SNR) at each sensor, where the noise is additive white Gaussian (AWGN).

•Estimate all 36 spherical harmonic coefficients for a plane wave sweeping over the entire 3D space and for all frequencies within the desired octave.

•We plot the real and imaginary parts of against the azimuth and elevation of the sweeping plane wave for lower, mid,and upper end of the frequency band.

Conclusions Spherical harmonic decomposition is a useful tool

to analyse 3D soundfields. Spherical arrays have inherent limitations that

make them unfeasible for practical implementation.

Circular microphone arrays and hybrid arrays need carefully designing based on underlying wave propagation and theory.

Combining circular arrays enables us to calculate odd and even harmonics independently, providing cleaner more accurate results.

Thanks&

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